Modeling subsurface heterogeneity is of crucial importance for the exploitation of subsurface resources and for the geological storage of nuclear wastes, CO2, etc. For more than half a century, geostatistics has been developed for this purpose and widely used in practice.
Geostatistics is the branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology, oceanography, geochemistry, geometallurgy, geography, forestry, environmental control, landscape ecology, soil science, and agriculture—especially in precision farming.
Geostatistics has gained popularity as a quantitative tool to generate multiple geological models, or “realizations,” that honor a given statistical structure and various types of measured and interpreted data. Two key approaches have traditionally been followed: a pixel based approach such as sequential indicator simulation based on variograms, and an object based (Boolean) approach in which large objects representing geobodies are inserted into the geological model. Both of these approaches are widely used, but have limitations in terms of reproducing realistic geobody shapes and conditioning to multiple types of data.
Multiple-point statistics (MPS) simulation is a more advanced spatial modeling technique because of its ability to reproduce complex geological patterns (e.g. sinuous channels) that cannot be modeled by two-point statistics (i.e., variograms). Unlike traditional geostatistics, MPS avoids the explicit definition of a random function, but directly infers the necessary multivariate distributions from “training images” or “TI”. This confers on MPS a potential applicability to any geological environment, provided that there is a training image representative of the geological heterogeneity and that the essential features of this training image can be characterized by statistics defined through a search template with a limited point configuration. The MPS approach appears flexible to data conditioning and to representing complex architectures of geological facies or petrophysical properties.
The goal of MPS approaches is to enable the ability to reproduce the geological “shapes” as object-based techniques do and maintain the data-conditioning flexibility as variogram-based techniques do. To achieve that, MPS needs to infer and reproduce multiple-point statistics way beyond the traditional two-point statistics (variogram). Because the available data are usually too sparse to infer such high-order statistics, Guardiano and Srivastava proposed the use of a training image, i.e. a three-dimensional numerical conceptual representation of the facies thought to be present in the reservoir to be modeled. The MPS simulation implementation is similar to Sequential Indicator Simulation, where the variogram is replaced with a training image, and Kriging is replaced with the following process to estimate local conditional facies probabilities:
1. Look for the n conditioning data (original well data or previously simulated values) closest to the grid node u to be simulated. These conditioning data form a data event dn that is fully characterized by its geometrical configuration (the data locations relative to u), and its data values (the facies at the data locations).
2. Scan the training image to find all replicates of dn (same geometric configuration and same data values as dn). For each replicate, record the facies value at the central location of the training replicate. By “central location,” what is meant is the grid node corresponding to the same relative location as u in the data event dn.
3. The estimated conditional probability of each facies at u is computed as the proportion of dn replicates that have this facies at their central locations.
One major advantage of this implementation is that, as in any pixel-based sequential simulation method, in contrast to object-based methods, well data are honored exactly. In addition, by capturing multiple-point statistics from the training image through the estimation of facies probabilities conditional to multiple-point data events, the MPS model reproduces training image patterns. However, the repetitive scanning of the training image to estimate facies probabilities is very time-consuming.
To help solve this issue, Strebelle introduced a dynamic data structure named “search tree” to store, prior to the simulation, all the conditional probability distributions that could be inferred from the training image. He also used a multiple-grid simulation approach that consists in simulating increasingly finer nested grids to capture training image patterns at various scales. In this multiple-grid approach, the conditioning data search neighborhood is defined by a template that only consists of nodes from the nested grid currently simulated. One search tree is built per data template, or per level of nested multiple grids.
The introduction of the search tree was the technical breakthrough that made SNESIM—a program developed by Strebelle—the first practical implementation of MPS simulation. Further progress was made a few years ago to improve MPS simulation time by reducing the size of the data template used to store the multi-point statistics in the search tree. The search template was designed such that it mostly consists of data locations corresponding to previously simulated nodes, i.e. nodes belonging to grids coarser than the grid currently simulated. Also, intermediary sub-grids were added to the traditional multiple-grid simulation approach to increase the relative proportion of previously simulated nodes in each nested grid.
These SNESIM implementation enhancements not only allowed users to reduce MPS simulation time, but also helped alleviate memory demand to build search trees. Alternative implementations of MPS simulation were proposed to tackle memory demand issues by classifying training patterns into a limited number of representative clusters, e.g., SIMPAT or FILTERSIM, but those solutions were found at the expense of increased simulation time and data conditioning issues.
The definition of the search template thus plays an important part for MPS simulations. It contains the maximum allowable conditioning data when simulating a cell. In theory, the larger the search template, the better the reproduction of the geometric features of the training image. But in practice, the size of the search template is limited by the CPU and RAM costs, and also by the size of the training image. For a given TI, a large search template may result in few replicates of patterns of that template size, and this leads to non-representative statistics. However, a small search template size will not be able to capture long-range patterns of the TI.
The use of multiple grids allows one to use a smaller size search template to save CPU time and at the same time to reproduce long-range patterns. The idea is to start the simulation on the cells at the coarsest level of the multiple grids and to finish it on the cells at the finest level of multiple grids. The search template geometry is defined once for all grids, but rescaled to each multiple grid level and a search tree is built for each multiple grid level.
The use of multiple grids is recommended, but a very large number of multiple grids will not necessarily improve the reproduction of long-range patterns because of the limited TI size that resulting in a limited number of data event replicates.
Thus, the ability to model long range patterns continues to present issues in fully realizing the benefits of MPS simulation.
Therefore, there is a need for improving MPS methods to capture long-range patterns or continuities without the concomitant enormous CPU and RAM costs.